STATISTICAL METHODS IN ECOMMERCE RESEARCH Chapter: Differential Equation Trees to Model Price Dynamics in Online Auctions

1.1 INTRODUCTION Empirical research of online auctions has been growing steadily in the last several years. Online auctions are different from offline auctions in several important ways: They are usually much longer, bidders and sellers are anonymous, and the barriers of entry are much lower for both bidders and sellers. These differences lead to auction dynamics that can be very different from those in offline auctions. One important aspect of these dynamics is their effect on the auction price. In this work, we are particularly interested in the price path of an online auction and its dynamics, that is, we are interested in the speed of the price increases and how it changes over the entire auction duration. Within the rich empirical online auction literature there has been very little in the way of studying the price dynamics. Most price-related studies have focused on the final price alone. However, in a series of recent papers by Jank & Shmueli and co-authors, they show that dynamics matter, that even auctions for the same product can have very different price paths and dynamics [9], and that incorporating the information contained in the price dynamics of an ongoing auction greatly improves the ability to forecast its final price [19]. In particular, [19] find that the availability of dynamics greatly improves the forecasting error compared to powerful competitors such as double exponential smoothing. Furthermore, the relationship between the price path and item properties) changes during the auction [2]. One example is the effect of the opening bid on the price at different times during the auctions. [2] and [18] find that although there is a positive relationship between the price and the opening bid at any point in the auction, the strength of this relationship declines as the auction progresses, implying that bidders derive less and less information from the opening bid. In order to estimate the price path and its dynamics from the discrete observed bids, Jank & Shmueli take a functional data analytic approach. In that approach, the price path of each auction is represented by a smooth continuous curve. The derivatives of this curve capture price dynamics: the first derivative captures the price velocity, indicating when the price increases fast and when the increase slows down. Similarly, the second derivative captures price acceleration. The estimation of smooth continuous price curves is achieved via smoothing methods, as is customary in functional data …